Abstract
We propose and discuss a model for flagellar mechanics in Euglena gracilis. We show that the peculiar nonplanar shapes of its beating flagellum, dubbed 'spinning lasso', arise from the mechanical interactions between two of its inner components, namely, the axoneme and the paraflagellar rod. The spontaneous shape of the axoneme and the resting shape of the paraflagellar rod are incompatible. Thus, the complex nonplanar configurations of the coupled system emerge as the energetically optimal compromise between the two antagonistic components. The model is able to reproduce the experimentally observed flagellar beats and the characteristic geometric signature of spinning lasso, namely, traveling waves of torsion with alternating sign along the length of the flagellum.
Introduction
Flagella and cilia propel swimming eukaryotic cells and drive fluids on epithelial tissues of higher organisms (Alberts et al., 2015). The inner structure of the eukaryotic flagellum is an arrangement of microtubules (MTs) and accessory proteins called the axoneme (Ax). A highly conserved structure in evolution, the Ax typically consists of nine cylindrically arranged MT doublets crossbridged by motor proteins of the dynein family. An internal central pair of MTs is connected by radial spokes to the nine peripheral doublets, determining the typical '9+2’ axonemal structure. Motor proteins hydrolyze ATP to generate forces that induce doublet sliding. Due to mechanical constraints exerted by linking proteins (nexins) and the basal body, dyneininduced sliding of MTs translates into bending movements of the whole structure. Motor proteins are thought to self regulate their activity via mechanical feedback, generating the periodic beats of flagella, see, for example, Brokaw, 2009 and Lindemann and Lesich, 2010.
Despite a general consensus on the existence of a selfregulatory mechanism, the inner working of the Ax is not fully understood and it is still the subject of active research (Wan and Jékely, 2020). While bendingthroughsliding is the accepted fundamental mechanism of flagellar motility, how specific flagellar shapes are determined is not yet clear. Nodal cilia present in early embryonic development display markedly nonplanar beats (Buceta et al., 2005). On the other hand, for the most studied swimming microorganisms, such as animal sperm cells and the biflagellate alga Chlamydomonas reinhardtii, the flagellar beat is, to a good approximation, planar. For these organisms, beat planarity is thought to be induced by the interdoublet links between one pair of MTs, typically those numbered 5 and 6 (Lin et al., 2012). These links inhibit the relative sliding of the 56 MTs pair, thus selecting a beating plane that passes through the center of the Ax and the midpoint between the inhibited MTs.
A remarkable deviation from the flagellar structure of the aforementioned organisms is found in euglenids and kinetoplastids. These flagellated protists have an extra element attached alongside the Ax (Cachon et al., 1988), a slender structure made of a latticelike arrangement of proteins called 'paraxial’ or 'paraflagellar’ rod (PFR), see Figure 1. The latter name is more common, but the former is possibly more accurate (Rosati et al., 1991). PFRs are attached via bonding links to up to four axonemal MTs, depending on the species (Walne and Dawson, 1993). PFRs are thought to be passive but, at least in the case of Euglena gracilis, some degree of activity is not completely ruled out (Piccinni, 1975).
E. gracilis has two flagella, designated as dorsal and ventral. The ventral flagellum remains within the reservoir, an invaginated region of the cell. The dorsal, PFRbearing, flagellum emerges from the reservoir and serves as a propulsive apparatus by means of periodic beating. In this paper, we show that the beating style of E. gracilis, sometimes dubbed ‘spinning lasso’ (Bovee, 1982), is characterized by a distinctive geometric signature, namely, traveling torsional peaks with alternating sign along the length of the flagellum. Moreover, we put forward and test the hypothesis that this distinctive beating style arises from the PFRAx mechanical interaction.
In order to put our hypothesis into context, we observe that the flagellar beat of PFRbearing kinetoplastid organisms, such as Leishmania and Crithidia, is planar (Gadelha et al., 2007). An apparent exception to beat planarity in kinetoplastids is found in the pathogenic parasite Trypanosoma brucei, which shows a characteristic nonplanar 'drilllike’ motion (Langousis and Hill, 2014). It has been claimed that the flagellar structure alone could account for the emergence this motion (Koyfman et al., 2011). However, the flagellum of T. brucei is not free, like that of Leishmania and Crithidia, but it is attached to the organism for most of its length, wrapped helically around the cell body. According to Alizadehrad et al., 2015, the flagellumbody mechanical interaction can alone explain T. brucei’s distinctive motion. Confirming this conclusion, Wheeler, 2017 showed that T. brucei mutants with bodydetached flagellum generate fairly planar beating. It is conjectured that the PFRAx bonds operate as the 56 interdoublet links in Chlamydomonas and sperm cells, inhibiting MTs sliding and selecting a plane of beat (Woolley et al., 2006).
The spinning lasso beat of E. gracilis does not conform to this scenario. Indeed, E. gracilis beating style is characterized by high asymmetry and nonplanarity. The full 3d flagellar kinematics of freely swimming cells has recently been revealed by Rossi et al., 2017 thanks to a mixed approach based on hydrodynamic theory and image analysis. As we report in the first part of this paper, the geometry of the spinning lasso is characterized by traveling waves of torsion with alternating sign along the flagellum length.
We argue that the key to the emergence of nonplanarity lies in a prominent structural asymmetry of the PFRAx attachment in euglenid flagella. Figure 1 shows a sketch of the crosssection of the euglenid flagellum redrawn from the electron microscopy studies by Melkonian et al., 1982 and Bouck et al., 1990. Following the latter studies, we number MTs in increasing order in the clockwise direction, as seen from the distal end of the Ax. Notice that a different convention is commonly used in structural studies of cilia and flagella, see, for example, Lin and Nicastro, 2018. The PFR is attached to MTs 1, 2, and 3. We consider two lines. One line (dashed) passes through the center of the Ax and MT 2, in the middle of the bonding complex. The other (solid) line connects the center of the Ax and the center of the PFR. The two lines cross each other. This is the structural feature on which we build our model.
In modeling the flagellar complex, we assume that the bonding links to the PFR select the local spontaneous beating plane of the Ax, from the same principle of MTs’ sliding inhibition discussed above. The local spontaneous beating plane so generated passes through the dashed line in Figure 1. We follow closely Hilfinger and Jülicher, 2008 and Sartori et al., 2016a in our modeling of the Ax, while we use a simple elastic spring model for the PFR. We show that, under generic actuation, the two flagellar components cannot be simultaneously in their respective states of minimal energy, and this crucially depends on the offset between the spontaneous beating plane of the Ax (dashed line in Figure 1) and the line joining the PFRAx centers (solid line in Figure 1). Instead, the typical outcome is an elastically frustrated configuration of the system, in which the two competing components drive each other out of plane. Under dyneins activation patterns that, in the absence of extraaxonemal structures, would produce an asymmetric beat similar to those of Chlamydomonas (Qin et al., 2015), or Volvox (Sareh et al., 2013), the model specifically predicts the torsional signature of the spinning lasso.
Interestingly, the lack of symmetry of the spinning lasso beat produces swimming trajectories with rotations coupled with translations (Rossi et al., 2017). In turn, cell body rotations have a key role in the lightguided navigation behavior of phototactic unicellular organisms (Goldstein, 2015), and of E. gracilis in particular, which has recently attracted renewed attention (Giometto et al., 2015; Ogawa et al., 2016; Tsang et al., 2018). Rotations along the major axis of the cell body produced by the spinning lasso beat allow the lightsensing apparatus of the organism to constantly scan the environment, and align E. gracilis with light intensity gradients.
In light of these observations, our analysis shows that the beat of the euglenid flagellum can be seen as an example of a biological function arising from the competition between antagonistic structural components. It is not dissimilar from the bodyflagellum interaction in T. brucei, which generates 3d motility. But the principle is much more general in biology and many other examples can be found across kingdoms and species, and at widely different scales. For instance in plants, a mechanism of seed dispersal arises from the mechanical competition between the two valves of the seed pods, see, for example Armon et al., 2011 for Bauhinia variegata and Hofhuis et al., 2016 for Cardamine hirsuta. Contraction by antagonistic muscles is key for animal movement and, in particular, for the functioning of hydrostatic skeletons (used from wormlike invertebrates to arms and tentacles of cephalopods, to the trunk of elephants, see Kier, 2012). The mechanical coupling of the helical periplasmic flagella to the rodshaped cell cylinder determines the flatwave morphology of the Lyme disease spirochete Borrelia burgdorferi (Dombrowski et al., 2009). Antagonistic contraction along perpendicularly oriented families of fibres is at work at the subcellular level, for example in the antagonistic motor protein dynamics in contractile ring structures important in eukaryotic cell division and development, see, for example Coffman et al., 2016. At the same subcellular scale, competing elastic forces arising from lipidprotein interactions are often crucial in determining the stability of complex shapes of the cellular membrane (Moser von Filseck et al., 2020), and in the case of the overall structure of the coronavirus envelope (Schoeman and Fielding, 2019).
Observations
We first analyze the experimental data from the 3d reconstruction of the beating euglenid flagellum obtained in Rossi et al., 2017 for freely swimming organisms. Swimming E. gracilis cells follow generalized helical trajectories coupled with rotation around the major axis of the cell body. It is precisely this rotation that allows for a 3d reconstruction of flagellar shapes from 2d videomicroscopy images. E. gracilis takes many beats to close one complete turn around its major axis. So, while rotating, cells expose their flagellar beat to the observer from many different sides. Stereomatching techniques can then be employed to reconstruct the flagellar beat in full (assuming periodicity and regularity of the beat). Figure 2 shows $N=10$ different curves in space describing the euglenid flagellum in different instants within a beat taken from Rossi et al., 2017. The reconstruction fits well experimental data from multiple specimens. The figure also illustrates the computed torsion of the flagellar curve at each instant (not previously published). Torsion, the rate of change of the binormal vector, is the geometric quantity that measures the deviation of a curve from a planar path (see the ‘Results’ Section below for the formal definition). The 'spinning lasso’ exhibits here a distinct torsional signature: torsion peaks of alternate sign that travel from the proximal to the distal end of the flagellum. We return to this point below.
To further investigate E. gracilis’ flagellar beat, we observed stationary cells trapped at the tip of a capillary. In this setting, the flagellum is not perturbed by the hydrodynamic forces associated with E. gracilis’ rototranslating swimming motion. The beat can then manifest itself in its most 'pristine’ form. We recorded trapped cells during periodic beating. We rotated the capillary and recorded the same beating cell from different viewpoints. Videomicroscopy images from one specimen are shown in Figure 3 and Video 1. While with fixed specimens we cannot reconstruct reliably the 3d flagellar shapes, Figure 3 shows that there is a high stereographical consistency with the flagellar shapes obtained from swimming organisms. Flagellar nonplanarity is thus not intrinsically associated with swimming, which reinforce the idea that the mechanism that generates nonplanar flagellar shapes might be structural in origin. Moreover, these observations justify the choice we made in our study to focus on a model of flagellar mechanics for stationary organisms, allowing for substantial simplifications.
Getting back to the torsion measurement in Figure 2, we show here that the pattern of torsional peaks of alternate sign is consistent with E. gracilis’ flagellar shapes as seen from common 2d microscopy, for either swimming or trapped organisms. Typically, the 2d outline (i.e. the projection on the focal plane) of a beating euglenid flagellum is that of a looping curve, see Figure 3 and, for example Tsang et al., 2018 for independent observations. Consider now an idealized 3d model of the spinning lasso geometry: a 'torsion dipole’. This simple geometric construction, shown in Figure 3, consists of a curve with two singular points of concentrated torsion with opposite sign. If we move along the curve, from proximal end to distal end, we first remain on a fixed plane (blue). Then the plane of the curve abruptly rotates by 90° (red plane) first, and then back by 90° in the opposite direction (yellow plane). These abrupt changes correspond to concentrated torsional peaks of opposite sign. When seen in a twodimensional projection, the torsion dipole generates a looping curve, which closely matches euglenid flagella’s outlines during a spinning lasso beat.
Mechanical model
We model Ax and PFR as cylindrical structures with deformable centerlines, see Figure 4. The euglenid flagellum is the composite structure consisting of Ax and PFR attached together. We suppose that the Ax is the only active component of the flagellum, whereas the PFR is purely passive. Our mechanical model builds on the definition of the total internal energy of the flagellum
which is given by the sum of three terms: the passive (elastic) internal energy ${\mathcal{W}}_{pas}^{a}$ of the Ax, the active internal energy ${\mathcal{W}}_{act}^{a}$ of the Ax (generated by dynein action), and the (passive, elastic) internal energy ${\mathcal{W}}^{p}$ of the PFR. The passive internal energy of the Ax is given by
where $U}_{1$ and $U}_{2$ are the bending strains of the Ax, $U}_{3$ is the twist, ${B}^{a}$ and ${C}^{a}$ are the bending and twist moduli (respectively), and $L$ is the total length of the Ax centerline ${\mathbf{\mathbf{r}}}^{a}$. Bending strains and twist depend on the arc length $s$ of the centerline, and they are defined as follows. We associate to the curve ${\mathbf{\mathbf{r}}}^{a}$ an orthonormal frame ${\mathbf{\mathbf{d}}}_{i}(s)$, with $i=1,2,3$, which determines the orientation of the orthogonal sections of the Ax (enclosed by light blue circles in Figure 4). The unit vectors ${\mathbf{\mathbf{d}}}_{1}(s)$ and ${\mathbf{\mathbf{d}}}_{2}(s)$ define the plane of the orthogonal section at s. The unit vector ${\mathbf{\mathbf{d}}}_{1}(s)$ lies on the line that connects the center of the Ax to MT 2, the center of the bonding links complex, see Figure 1. The unit vector ${\mathbf{\mathbf{d}}}_{3}(s)={\partial}_{s}{\mathbf{\mathbf{r}}}^{a}(s)$ lies perpendicular to the orthogonal section. Bending strains and twist are then given by
Thus, $U}_{1$ and $U}_{2$ measure the bending of the Ax on the local planes ${\mathbf{\mathbf{d}}}_{2}$${\mathbf{\mathbf{d}}}_{3}$ and ${\mathbf{\mathbf{d}}}_{3}$${\mathbf{\mathbf{d}}}_{1}$, respectively, while the twist $U}_{3$ is given by the rotation rate of the orthonormal frame around the tangent ${\mathbf{\mathbf{d}}}_{3}$ to the centerline.
We remark that the righthand side of Equation 2 is formally identical to a classical expression arising in Kirchhoff’s theory for elastic rods (Goriely, 2017). Our rod is however nonstandard because it consists of a hollow tubular structure arising as the envelope of nine individual MTs. In Appendix 1, we model each of the MTs as a standard rod with (full crosssection and) bending and twisting moduli ${B}^{m}$ and ${C}^{m}$. We then show that the geometry and deformations of the Ax (centerline ${\mathbf{\mathbf{r}}}^{a}$, frame vectors ${\mathbf{\mathbf{d}}}_{1}$, ${\mathbf{\mathbf{d}}}_{2}$, and ${\mathbf{\mathbf{d}}}_{3}$, bending strains and twist) determine the geometry and deformations of the individual MTs. By summing the elastic contributions of individual MTs, we obtain Equation 2 as the elastic energy of the assembly, with ${B}^{a}=9{B}^{m}$ and ${C}^{a}=9{C}^{m}$.
The active internal energy of the Ax is defined as minus the total mechanical work of the dyneins
where ${\gamma}_{1}$ and ${\gamma}_{2}$ are the two variables that quantify the shear (i.e., collective sliding) of MTs, while $H}_{1$ and $H}_{2$ are the corresponding shear forces exerted by molecular motors. Following Sartori et al., 2016b we also allow for singular shear forces, ${\widehat{H}}_{1}$ and ${\widehat{H}}_{2}$, concentrated at the distal end of the Ax. These forces arise naturally, as we remark after Equation 23 in the ‘Results’ Section.
The active internal energy can be written in a more natural way at the level of individual MT pairs in terms of the work done by the sliding forces ${F}_{j}$ generated by the dyneins crossbridging MTs $j$ and $j+1$ against their relative sliding displacements ${\sigma}_{j}$, for $j=1,2,\mathrm{\dots},9$. These forces and displacements are defined in detail in Appendix 1, and their work computed in Equation 35. The way Equation 35 gives rise to the equivalent reformulation (Equation 4) in terms of global crosssection variables, the forces ${H}_{i}$ and shears ${\gamma}_{i}$, is also discussed there. Here, we simply notice that the structural constraints of the Ax lead to simplifications on the kinematics. These constraints do not allow MTs to slide by whatever amount, and the sliding of MT pairs ${\sigma}_{j}$ are not independent. Rather, there are only two degrees of freedom that determine MTs sliding, which are given by the shear variables ${\gamma}_{1}$ and ${\gamma}_{2}$. Moreover, again due to the structural constraints of the Ax, the shear variables are coupled to the bending strains (Equation 3), as discussed further below. In Appendix 1, we derive the linear relations between the shear variables and the individual sliding of MT pairs (Equation 36). We also compute the relations between the dynein forces ${F}_{j}$ acting on each pair of adjacent MTs and the shear forces $H}_{1$ and $H}_{2$ (Equation 37). The singular shear forces ${\widehat{H}}_{1}$ and ${\widehat{H}}_{2}$ arise from concentrated sliding forces ${\widehat{F}}_{j}$ at the distal end of the Ax in an analogous way.
To explain how the shear variables ${\gamma}_{1}$ and ${\gamma}_{2}$ are related to the MTs' kinematics, we observe that the MTs centerlines ${\mathbf{\mathbf{r}}}^{j}$, for $j=1,2\mathrm{\dots}9$ , are given by ${\mathbf{\mathbf{r}}}^{j}(s)=\mathbf{\mathbf{C}}(s,{\varphi}_{j})$, where ${\varphi}_{j}=2\pi (2j)/9$, and
is the parametrization of the cylindrical surface of the Ax (${\rho}^{a}$ is the Ax radius) in terms of the centerline arc length $s$ and the angle $\varphi $. A special axonemal deformation with ${\gamma}_{2}=0$ is shown in Figure 4. In this case, the Ax is bent into a circular arc, and the centerline ${\mathbf{\mathbf{r}}}^{a}$ lies on the plane generated by the unit vectors ${\mathbf{\mathbf{d}}}_{1}$ and ${\mathbf{\mathbf{d}}}_{3}$. The shear variable ${\gamma}_{1}(s)\ne 0$ has here a simple geometrical interpretation. For each fixed $s$ the curve $\varphi \mapsto \mathbf{\mathbf{C}}(s,\varphi )$ describes what we call the ‘material’ section of the Ax at $s$ (red curves in Figure 4). The material section is a planar ellipse centered in ${\mathbf{\mathbf{r}}}^{a}(s)$ which connects points of neighboring MTs’ corresponding to the same arc length. Equation 5 says that ${\gamma}_{1}(s)$ is the tangent of the angle at which the material sections at $s$ intersect the orthogonal sections at $s$.
As mentioned above, the kinematic constraints of the Ax couple the shear variables with bending strains. We have
The above formulas (whose detailed derivation is given in Appendix 1) underlie the essential mechanism of axonemal motility: collective sliding of MTs generates bending of the whole Ax. We point out here that there is no coupling between the shear variables ${\gamma}_{1},{\gamma}_{2}$ and the twist $U}_{3$, a fact that will have consequences in the remainder.
The special axonemal deformation in Figure 4 shows the case in which ${U}_{1}(s)=0$ and ${U}_{2}(s)=K$, so that the Ax is bent into a circular arc of radius $1/K$. While ${\gamma}_{2}(s)=0$, the shear variable ${\gamma}_{1}(s)=Ks$ increases linearly with $s$. Material and orthogonal sections coincide at the base (the basal body impose no shear at $s=0$) and the angle between them grows as we move along the centerline towards the distal end of the Ax. In order for the Ax to bend, MTs from one side of the Ax must be driven toward the distal end while the others must be driven toward the proximal end.
We remark here that Equation 4 defines the most general active internal energy generated by molecular motors, and we do not assume at this stage any specific (spatial) organization of dynein forces. We will introduce specific shear forces later in the ‘Results’ Section.
The PFR is modelled as an elastic cylinder with circular cross sections of radius ${\rho}^{p}$ and rest length $L$. We assume that the PFR can stretch and shear. The total internal energy of the PFR is given by
where $V}_{1$ and $V}_{2$ are the shear strains, $V}_{3$ is the stretch, ${D}^{p}$ and ${E}^{p}$ are the shear and stretching moduli, respectively. We are neglecting here the PFR’s bending and twisting stiffness. Classical estimations on homogeneous elastic rods, see, for example Goriely, 2017, show that bending and twist moduli scale with the fourth power of the cross section radius, whereas shear and stretching moduli scale with the second power and hence they are dominant for small radii. We assume that dynein forces are strong enough to induce shear in the PFR, thus PFR’s bending and twist contributions to the energy of the flagellum become negligible. We are also neglecting Poisson effects by treating the PFR crosssections as rigid.
The PFR shear strains and stretch are defined as follows. The crosssections centers of the PFR lie on the curve ${\mathbf{\mathbf{r}}}^{d}$, and their orientations are given by the orthonormal frame ${\mathbf{\mathbf{g}}}_{i}(s)$, with $i=1,2,3$. The unit vectors ${\mathbf{\mathbf{g}}}_{1}(s)$ and ${\mathbf{\mathbf{g}}}_{2}(s)$ determine the cross section plane centered at ${\mathbf{\mathbf{r}}}^{p}(s)$, while the unit vector ${\mathbf{\mathbf{g}}}_{3}(s)$ is orthogonal to it. The curve ${\mathbf{\mathbf{r}}}^{p}$ is not parametrized by arc length and ${\mathbf{\mathbf{g}}}_{3}$ is not in general aligned with the tangent to ${\mathbf{\mathbf{r}}}^{p}$. Shear strains and stretch are given by the formulas
The shear strains thus depend on the orientation of the cross sections with respect to the centerline (tangent), while the stretch measures the elongation of the centerline.
The PFRAx attachment couples the kinematics of the two substructures, see Figure 4. In the remainder, we formalize the attachment constraint and we show how the PFR’s shear strains and stretch (Equation 8), and thus the flagellar energy (Equation 1), are completely determined by the Ax kinematic variables.
For each $s$, the PFR crosssection centered at ${\mathbf{\mathbf{r}}}^{p}(s)$ is in contact with the Ax surface at the point $\mathbf{\mathbf{C}}(s,{\varphi}^{p})$ for a fixed angle coordinate ${\varphi}^{p}$, see Figure 1 and Melkonian et al., 1982. The PFR centerline is given by
where $\mathbf{\mathbf{N}}(s,{\varphi}^{p})\approx {\mathbf{\mathbf{d}}}_{1}(s)\mathrm{cos}{\varphi}^{p}+{\mathbf{\mathbf{d}}}_{2}(s)\mathrm{sin}{\varphi}^{p}$ is the outer unit normal to the axonemal surface at $\mathbf{\mathbf{C}}(s,{\varphi}^{p})$. The normal vector $\mathbf{\mathbf{N}}(s,{\varphi}^{p})$ lies on the plane of the PFR crosssection centered at ${\mathbf{\mathbf{r}}}^{p}(s)$. Indeed, we have ${\mathbf{\mathbf{g}}}_{1}(s)=\mathbf{\mathbf{N}}(s,{\varphi}^{p})$ for the first unit vector of the PFR orthonormal frame. Only one more degree of freedom remains, namely ${\mathbf{\mathbf{g}}}_{2}(s)$, which must be orthogonal to $\mathbf{\mathbf{N}}(s,{\varphi}^{p})$, to fully characterize the orientations of the PFR crosssections. Here is where the bonding links attachments are introduced in the model. The bonding links of the PFR crosssection centered at ${\mathbf{\mathbf{r}}}^{p}(s)$ are attached to three adjacent MTs at the same MTs’ arc length $s$. The individual attachments are therefore located on the material section of the Ax at $s$. Given this, ${\mathbf{\mathbf{g}}}_{2}(s)$ is imposed to be parallel to ${\partial}_{\varphi}\mathbf{\mathbf{C}}(s,{\varphi}^{p})$, the tangent vector to the material section of the Ax at the point of contact $\mathbf{\mathbf{C}}(s,{\varphi}^{p})$, see Figure 4. This condition critically couples MTs’ shear to the orientations of the PFR crosssections, as further demonstrated below.
To summarize, we have the following formulas for the PFR orthonormal frame vectors
By replacing the expressions in Equations 910 in Equation 8, we obtain formulas for the shear strains and stretch of the PFR in terms of the Ax kinematic parameters. The shear strain $V}_{1$ and the stretch $V}_{3$ are found to be of order ${\rho}^{p}\sim {\rho}^{a}$ (see Appendix 1 for detailed calculations). Since ${\rho}^{p}$ is small compared to the length scale $L$ of both PFR and Ax, we neglect these quantities. The only nonnegligible contribution to the PFR energy is thus given by the shear strain $V}_{2$. After linearization, we have ${V}_{2}\approx \mathrm{sin}{\varphi}^{p}{\gamma}_{1}+\mathrm{cos}{\varphi}^{p}{\gamma}_{2}$. The PFR energy in terms of Ax kinematic parameters is then given by
The shear of axonemal MTs determines the orientation of the PFR crosssections. In Figure 5 (middle pictures), we show an example of this kinematic interplay. The Ax is again bent in an arc of a circle on the plane ${\mathbf{\mathbf{d}}}_{1}{\mathbf{\mathbf{d}}}_{3}$, with ${U}_{1}(s)=0$, ${U}_{2}(s)=K$, ${\gamma}_{1}(s)=Ks$, and ${\gamma}_{2}(s)=0$. PFR and Ax centerlines run parallel to each other, indeed from Equation 9 we have that ${\partial}_{s}{\mathbf{\mathbf{r}}}^{a}\approx {\partial}_{s}{\mathbf{\mathbf{r}}}^{p}$ for every deformation. The linking bonds impose a rotation of the cross sections of the PFR as we progress from the proximal to the distal end of the flagellum, generating shear strain ${V}_{2}(s)=\mathrm{sin}{\varphi}^{p}{\gamma}_{1}(s)=\mathrm{sin}{\varphi}^{p}Ks$ on the PFR. This mechanical interplay leads to nonplanarity of the euglenid flagellar beat. This mechanism is controlled by the offset between the PFRAx joining line and the local spontaneous bending plane of the Ax, as further discussed in the ‘Results’ Section.
Equilibria
Under generic (steady) dynein actuation, that is, given $H}_{1$ and $H}_{2$ (not timedependent), and in the absence of external forces, the flagellum deforms to its equilibrium configuration $\delta \mathcal{W}=0$. Bending strains and twist at equilibrium solve the equations
where $\mathbf{\mathbf{U}}=({U}_{1},{U}_{2})$ is the bending vector, ${\mathbf{\mathbf{e}}}_{p}=(\mathrm{cos}{\varphi}^{p},\mathrm{sin}{\varphi}^{p})$, and ${\mathbf{\mathbf{H}}}^{\perp}=({H}_{2},{H}_{1})$. We use the symbol $\mathbf{\mathbf{a}}\otimes \mathbf{\mathbf{b}}$ to denote the matrix with components ${(\mathbf{\mathbf{a}}\otimes \mathbf{\mathbf{b}})}_{ij}={a}_{i}{b}_{j}$. The field Equation 12 is complemented by the boundary conditions
where ${\widehat{\mathbf{\mathbf{H}}}}^{\perp}=({\widehat{H}}_{2},{\widehat{H}}_{1})$. Equations 12, 13 can be interpreted as the torque balance equations of the Ax. The derivative of the (elastic) bending moment and the internal shear stresses balance the torque per unit length exerted by the PFR on the Ax, which is given by the ${D}^{p}$dependent term appearing in the first equation. The torque depends on the integral of the bending vector, making the balance equations nonstandard (integrodifferental instead of differential). This dependency is due to the fact that the torque arises from the shear deformations of the PFR, which are induced by the shear of axonemal MTs, which is, in turn, related to axonemal bending strains via the integral relations (Equation 6). The torque exerted by the PFR on the Ax is sensitive to the direction given by the unit vector ${\mathbf{\mathbf{e}}}_{p}$, hence it depends on the angle ${\varphi}^{p}$ between the AxPFR joining line and the unit vector ${\mathbf{\mathbf{d}}}_{1}$.
Hydrodynamics
We consider here our mechanical model in the presence of external forces. For simplicity, we ignore the possible forces exerted by the cell surface on the nonemergent portion of the flagellum, located inside the reservoir of the cell. Indeed, we suppose that the flagellum is immotile and straight in the region inside the reservoir. We can assume, therefore, that our model effectively describes the flagellum from its emergence point outwards. The only forces the flagellum is subject to come from fluid interaction, which we assume to act all along its length. We consider the extended functional
where $\mathbf{\mathbf{\Lambda}}$ is the Lagrange multiplier vector enforcing the constraint ${\partial}_{s}{\mathbf{\mathbf{r}}}^{a}={\mathbf{\mathbf{d}}}_{3}$. We treat the fluidflagellum interaction in the local drag approximation of Resistive Force Theory, see for example Wolgemuth et al., 2000. In this approximation, viscous forces and torques depend locally on the translational and rotational velocity of the flagellum, represented here for simplicity by the translational and rotational velocity of the Ax. The external viscous forces $\mathbf{\mathbf{F}}$ and torques $\mathbf{\mathbf{G}}$ (per unit length) acting on the flagellum are given by
where ${\mu}_{\perp}$, ${\mu}_{}$, and ${\mu}_{r}$ are the normal, parallel, and rotational drag coefficient (respectively), and $\mathbf{\mathbf{I}\mathbf{d}}$ is the identity tensor. The principle of virtual work imposes
for every variation $\delta {\mathbf{\mathbf{r}}}^{a}$ and $\delta \mathit{\bm{\theta}}=\delta {\theta}_{1}{\mathbf{\mathbf{d}}}_{1}+\delta {\theta}_{2}{\mathbf{\mathbf{d}}}_{2}+\delta {\theta}_{3}{\mathbf{\mathbf{d}}}_{3}$, where $\delta {\theta}_{1}=(\delta {\mathbf{\mathbf{d}}}_{2}\cdot {\mathbf{\mathbf{d}}}_{3})$, $\delta {\theta}_{2}=(\delta {\mathbf{\mathbf{d}}}_{3}\cdot {\mathbf{\mathbf{d}}}_{1})$, and $\delta {\theta}_{3}=(\delta {\mathbf{\mathbf{d}}}_{1}\cdot {\mathbf{\mathbf{d}}}_{2})$. Linearizing the force balance equations derived from Equation 16 we obtain the following equations for bending strains and twist
which are decoupled from the extra unknown $\mathbf{\mathbf{\Lambda}}$. Equations 17 and 18 are complemented by the boundary conditions
The details of the derivation of Equations 17–20 are provided in Appendix 2.
Once we solve for $U}_{1$, $U}_{2$, and $U}_{3$ either the equilibrium Equations 12, 13 or the dynamic Equations 17–20, the shape of the flagellum can be recovered. In particular, we obtain the orthonormal frame ${\mathbf{\mathbf{d}}}_{i}$ with $i=1,2,3$ by solving Equation 53, while the centerline of the Ax is recovered by integrating ${\partial}_{s}{\mathbf{\mathbf{r}}}^{a}={\mathbf{\mathbf{d}}}_{3}$.
Results
We analyze the geometry of the centerline ${\mathbf{\mathbf{r}}}^{a}$ which, due to the slenderness of the flagellar structure, is a close proxy for the shape of the flagellum.
In general, the shape of a curve is determined by its curvature $\kappa$ and torsion $\tau$. Since ${\mathbf{\mathbf{r}}}^{a}$ is parametrized by arc length, the two quantities are given by the formulas $\mathrm{\partial}}_{s}\mathbf{t}=\kappa \mathbf{n$ and ${\partial}_{s}\mathbf{\mathbf{b}}=\tau \mathbf{\mathbf{n}}$, where $\mathbf{\mathbf{t}}={\partial}_{s}{\mathbf{\mathbf{r}}}^{a}$, $\mathbf{\mathbf{n}}={\partial}_{s}\mathbf{\mathbf{t}}/{\partial}_{s}\mathbf{\mathbf{t}}$, and $\mathbf{\mathbf{b}}=\mathbf{\mathbf{t}}\times \mathbf{\mathbf{n}}$ are the tangent, normal, and binormal vector to the curve ${\mathbf{\mathbf{r}}}^{a}$, respectively. Given $\kappa$ and $\tau$, ${\mathbf{\mathbf{r}}}^{a}$ is uniquely determined up to rigid motions.
From the previous definitions and from Equation 3 we obtain the relations between curvature, torsion, bending, and twist. In compact form these relations are given by
which hold for $\mathbf{\mathbf{U}}\ne 0$. In Equation 21 we introduced the angle $\psi$ that the bending vector $\mathbf{\mathbf{U}}=({U}_{1},{U}_{2})$ forms with the line ${U}_{2}=0$, see Figure 6. Now, at equilibrium (Equation 12) we have
under any dynein actuation. In other words, axonemal deformations are twistless. This is, fundamentally, a consequence of the fact that shear of axonemal MTs and twist are uncoupled (Equation 6). The torsion of the centerline ${\mathbf{\mathbf{r}}}^{a}$ is our main focus, since we are interested in emergent nonplanarity. Combining Equation 21 and Equation 22 we have that torsion can arise only from the rotation rate ${\partial}_{s}\psi $ of the bending vector $\mathbf{\mathbf{U}}$ along the length of the flagellum. This last observation will be important in the following.
Dyneins’ actuation induced by sliding inhibition
Under the assumptions Equation 22 and Equation 24, the flagellar energy (Equation 1) can be rewritten as
are the target bending strains generated by the dynein forces. The use of this terminology is clear from Equation 23. The effect of dynein actuation at equilibrium (when the energy is minimized) is to bring the bending strains $U}_{1$ and $U}_{2$ as close as possible to ${U}_{1}^{*}$ and ${U}_{2}^{*}$, respectively. The emerging bending strains and the target bending strains might not match due to the interference by the PFR component of the energy (${D}^{p}\ne 0$). From the formulas for the target bending strains in Equation 23, we can infer the importance of the concentrated shear forces ${\widehat{H}}_{1}$ and ${\widehat{H}}_{2}$. Without these forces, admissible spontaneous configurations of the Ax would be ruled out. If the concentrated shear forces are null, for example, the Ax cannot spontaneously bend into a circular arc. Indeed, for a circular arc of radius $1/K$ on the plane ${\mathbf{\mathbf{d}}}_{1}$${\mathbf{\mathbf{d}}}_{3}$ we must have ${U}_{2}^{*}=0$ and ${U}_{2}^{*}=K$. In this case, from Equation 23 we have that ${H}_{1}/{B}^{a}={\partial}_{s}{U}_{2}^{*}=0$, which implies ${H}_{1}=0$ and ${\widehat{H}}_{1}={B}^{a}K$, so the concentrated forces must be non null.
Our working hypothesis is that MTs' sliding stretches the AxPFR bonding links, which, in turn, inhibits sliding of MTs’ 1, 2, and 3, and triggers a dynein organization (via mechanical feedback) similar to the one present in Chlamydomonas. We take this feedbackbased selforganization process as a given, and we consider a force pattern that produces local spontaneous bending on the plane ${\mathbf{\mathbf{d}}}_{1}(s)$${\mathbf{\mathbf{d}}}_{3}(s)$, as shown in Figure 1. This is equivalent to require that ${U}_{1}^{*}=0$, which leads to the following condition on the shear forces
Emergence of nonplanarity
We consider here the equilibrium Equation 12 under the hypothesis (Equation 24). We look at the equilibrium configurations for every possible value of the angle ${\varphi}^{p}$ between the AxPFR joining line and the spontaneous bending plane of the Ax, even though the value of actual interest for E. gracilis is ${\varphi}^{p}\approx 2\pi /9$. We can prove analytically the following statement: if the AxPFR joining line is neither parallel nor orthogonal to the spontaneous bending plane of the Ax, then the emergent flagellar shapes are nonplanar.
Indeed, suppose ${H}_{1}\ne 0$. From Equation 21 and Equation 22 it follows that the shape of the flagellum is planar $(\tau =0)$ if and only if the angle $\psi$ of the bending vector $\mathbf{\mathbf{U}}(s)=({U}_{1}(s),{U}_{2}(s))$ is constant. The bending vector must therefore be confined on a line for every $s$. In this case there must be two constants $c}_{1$ and $c}_{2$ such that ${U}_{1}(s)={c}_{1}U(s)$ and ${U}_{2}(s)={c}_{2}U(s)$ for some scalar function $U$. Now, if a planar $\mathbf{\mathbf{U}}$ is a solution of Equation 12, we must have
with ${c}_{1}U(L)=0$ and ${c}_{2}U(L)={\widehat{H}}_{1}/{B}^{a}$. If ${\varphi}^{p}\notin \{0,\pi /2,\pi ,3\pi /2\}$ the system of Equations 25, 26 admits no solution. Indeed, suppose first that ${\widehat{H}}_{1}=0$. Since ${H}_{1}\ne 0$ we must have $({c}_{1},{c}_{2})\ne (0,0)$. However, in this case, Equation 25 admits the unique solution $U=0$, which is incompatible with Equation 26. If ${\widehat{H}}_{1}\ne 0$, then the boundary conditions impose ${c}_{1}=0$, but in this case Equation 25 has again $U=0$ as a unique solution, which is incompatible with both the boundary conditions and with Equation 26. Our statement is thus proved.
For ${\varphi}^{p}\approx 2\pi /9$, the characteristic value for E. gracilis, the nonplanarity of flagellar shapes is not just possible. It is the only outcome under any nontrivial dynein actuation.
Structural incompatibility and torsion with alternating sign
Alongside the previous analysis, there is a less technical way to infer the emergence of nonplanarity from our model. We look here more closely to the flagellum energy, and we think in terms of structural incompatibility between Ax and PFR, seen as antagonistic elements of the flagellum assembly, see Figure 5.
Under the assumptions Equation 22 and Equation 24, the flagellar energy is given by
with ${U}_{2}^{*}$ as in Equation 23. The energy has two components, ${\mathcal{W}}^{a}$ that depends on the Ax bending modulus ${B}^{a}$, and ${\mathcal{W}}^{p}$ that depends on the PFR shear modulus ${D}^{p}$. We can vary these material parameters and explore what the resulting minima of $\mathcal{W}$, that is, the equilibrium configurations (Equations 1213), must look like. We consider the nondimensional parameter $\nu ={D}^{p}/({B}^{a}{L}^{2})$. When $\nu \ll 1$ the Ax component ${\mathcal{W}}^{a}$ of the energy dominates. In this case, at equilibrium, the bending vector has to be close to the target bending vector $\mathbf{\mathbf{U}}\approx (0,{U}_{2}^{*})$. In particular, then, $\mathbf{\mathbf{U}}(s)$ will be confined near the line ${U}_{1}=0$ for every $s$. In the case $\nu \gg 1$ the PFR component ${\mathcal{W}}^{p}$ dominates, and the energy is minimized when $\mathbf{\mathbf{U}}(s)$ lies close to the line generated by the vector ${\mathbf{\mathbf{e}}}_{p}^{\perp}=(\mathrm{sin}{\varphi}_{p},\mathrm{cos}{\varphi}_{p})$. Clearly, if the latter line is different from ${U}_{1}=0$, the two extreme regimes $\nu \ll 1$ and $\nu \gg 1$, each of which favours one of the two individual components, aim at two different equilibrium configurations. In other words, Ax and PFR are structurally incompatible.
When neither of the two energy components dominates, the emergence of nonplanarity can be intuitively predicted with the following reasoning. In the intermediate case of $\nu \sim 1$, we expect the equilibrium configurations to be a compromise among the two extreme cases, with the bending vector $\mathbf{\mathbf{U}}(s)$ being ‘spread out’ in the region between the two extreme equilibrium lines. The spreading of the bending vector is aided by the concentrated shear force at the tip, which imposes $\mathbf{\mathbf{U}}(L)=(0,{U}_{2}^{*}(L))$ irrespectively of the PFR stiffness. The bending vector is then ‘pinned’ at $s=L$ on the ${U}_{1}=0$ line while it gets dragged toward the line generated by ${\mathbf{\mathbf{e}}}_{p}^{\perp}$ for large values of $\nu$. Hence the spreading. The bending vector will then span an area and, consequently, undergo rotations. Since torsion is determined by the rotation rate of the bending vector ($\tau ={\partial}_{s}\psi $), the resulting flagellar shapes will be nonplanar.
Figure 6 illustrates a critical example in which the previous intuitive reasoning effectively plays out. We consider a target bending of the kind ${U}_{2}^{*}(s)={A}_{0}+{A}_{1}\mathrm{sin}(2\pi s/L)$, a fair idealization of the asymmetric shapes of a Chlamydomonaslike flagellar beat (Geyer et al., 2016). We take ${\varphi}^{p}=\pi /4$ (larger than the E. gracilis value, to obtain clearer graphs). For $\nu =0$, the bending vector lies inside the ${U}_{1}=0$ line, and its amplitude oscillates. For positive values of $\nu$, when the PFR stiffness is 'turned on’, the oscillating bending vector is extruded from the ${U}_{1}=0$ line. For large values of $\nu$ it gets closer and closer to the line generated by ${\mathbf{\mathbf{e}}}_{p}^{\perp}$. The bending vector spans an area and, following the oscillations, it rotates clockwise and anticlockwise generating an alternation in the torsion sign. This is the geometric signature of the spinning lasso.
In Appendix 3, we relaxed the planar constraint (Equation 24), and considered different weakly nonplanar spontaneous configurations, by perturbing Chlamydomonaslike target bending strains. Each configuration presents a different (nonnull) torsional profile when the antagonistic mechanical interaction of the PFR is absent $\nu =0$. For larger values of $\nu$, each perturbation assumes the typical spinning lasso geometry (torsional peaks of alternate sign), regardless of the native ($\nu =0$) torsional profile. This shows that the PFRAx interactions strongly influence the flagellar shape outcome even when the perfectly planar constraint on the spontaneous bending is relaxed.
Hydrodynamic simulations and comparison with observations
Our model is able to predict the torsional characteristic of the euglenid flagellum in the static case, and in absence of external forces. We test here the model in the more realistic setting of timedependent dynein actuation in the presence of hydrodynamic interactions.
We first observe that, as in the static case, the dynamic equations for $\mathbf{\mathbf{U}}$ and $U}_{3$ are decoupled (Equations 17, 18), and that dynein forces do not affect twist. We have then twistless kinematics under any actuation also in the dynamic case, at least after a time transient. We can simply assume Equation 22 for all times, so the torsion of ${\mathbf{\mathbf{r}}}^{a}$ is still completely determined by the bending vector.
We consider a dynein actuation that generates Chlamydomonaslike shapes in a flagellum with no extraaxonemal structures. The shear forces $H}_{1$ and ${\widehat{H}}_{1}$ employed in our simulation are shown in Figure 7. The same figure also shows the emergent bending strains of a PFRfree flagellum actuated by said forces, beating in a viscous fluid. The dynamic equations for this system are simply Equations 17, 18 with ${D}^{p}=0$. The resulting bending strains, which generate a planar beat, resembles the experimentally observed Chlamydomonas flagellar curvatures reported in Qin et al., 2015.
Finally, Figure 7 presents the emergent bending strains of the beating euglenid (PFRbearing) flagellum, together with the corresponding flagellar torsion. The spinning lasso torsional signature is clearly present. Indeed, the fluidstructure interaction does not disrupt the AxPFR structural incompatibility, which still generates nonplanar shapes with traveling waves of torsional peaks with alternating sign and the typical loopingcurve outlines, Figure 2. All the details on the methods and parameters employed in the simulations are given in Appendix 4. Video 2 shows a comparison between the simulated flagellar beat and the experimental observations.
Swimming simulations
The unique flagellar beat of E. gracilis is at the base of the distinctive behavior of the organism, producing the typical rototranslational trajectories of swimming cells. This has been demonstrated by swimming simulations using the experimentally measured flagellar shapes in Rossi et al., 2017 with a Resistive Force Theory approach to model hydrodynamic interactions between the cell and the surrounding fluid. Similar conclusions are reached with a Boundary Element Method for the computation of the fluid flows induced by the same measured history of the flagellar shapes, see Giuliani et al., 2021. We carried out here analogous simulations using the theoretical waveforms produced by our model.
The application of Resistive Force Theory hydrodynamics to the waveforms shown in Figure 7 produces swimming paths similar to the typical observed trajectories. Swimming cell simulations are reported in Figure 8, see also Video 3. The cell is propelled by the flagellum, following a generalized righthanded helical trajectory while rotating around its major axis. After each full turn of the helix the cell completes one full body rotation. For more details on the implementation and physics behind swimming simulation see Appendix 5.
Discussion and outlook
We have shown how the origin of the peculiar shapes of the euglenid flagellum can be explained by the mechanical interplay of two antagonistic flagellar components, the Ax and the PFR. Our conclusions are based mainly on the hypothesis that sliding inhibition by the PFR organizes dynein activity, and localizes the spontaneous bending plane of the Ax as the one that passes from the Ax center through the MTs bonded to the PFR. This is in agreement with the current understanding of the mechanism that generates beat planarity in other PFRbearing flagellar systems. Nonplanarity in E. gracilis can arise because of a marked asymmetry in the Axbonding linksPFR complex in the euglenid flagellum, which is not found in kinetoplastids such as Leishmania (Gluenz et al., 2010) or Trypanosoma (Portman and Gull, 2010).
In the absence of a precise knowledge of the dynein actuation pattern, we tested our mechanical model under shear forces that would, in the absence of extraaxonemal structures, realize a beat similar to those found in model systems like Chlamydomonas. We appreciate that the emergent distortion of the Ax, generated by the AxPFR interplay, could in principle lead to different actuation patterns, consistently with the hypothesis of dynein actuation via mechanical feedback. Including dynein feedback in the euglenid flagellum model we proposed will require further study, and can potentially open new avenues for the study of ciliary motility in general. While the existence of a mechanical feedback between molecular motors and the flagellar scaffold is fairly accepted, there are several competing theories arguing in favour of different feedback mechanisms. The structural and kinematic peculiarities of the E. gracilis beat may provide a challenging new model system to test the relative merits of these alternative theories.
Along with the mechanism that let the euglenid flagellar shapes emerge, it is worth considering how this characteristic flagellar beat is integrated in the overall behavior of the organism. The spinning lasso produces the typical rototranslational motion of E. gracilis cells. Cell body rotation is in turn associated with phototaxis. Indeed, rotation allows cells to veer to the light source direction when stimulated, or escape in the opposite direction, when the signal is too strong. Here, the key biochemical mechanism could be the one often found in nature, by which periodic signals generated by lighting and shading associated with body rotations are used for navigation, in the sense that the existence of periodicity implies a lack of proper alignment (Goldstein, 2015). It is known that the PFR is directly connected with the lightsensing apparatus (Rosati et al., 1991), and might even be contractile (Piccinni, 1975). Transient light stimuli have been shown to change flagellar beat patterns. While the euglenid flagellum consistently present looping outlines (the 2d trace of a torsion dipole), its extension from the cell body changes with changing light intensity signals, resulting in a variety of 'quantized’ phototaxis behaviors (Tsang et al., 2018). With suitable modifications, our model could provide a starting point to address the (possible) mechanisms of active contraction of the PFR, light perception, and their interplay. Further study on euglenid flagellar motility and phototaxis could lead to a more comprehensive understanding of the biomechanical role of PFR, both in phototaxis and in general.
The very basic question of 'why' euglenids have evolved a structure such as PFR is an interesting challenge, but beyond the scope of the model we have developed in this paper. Our results confirm, however, the interest of euglenids as model systems for responsive unicellular organisms: E. gracilis, in particular, is a unicellular organism exhibiting a variety of motility behaviors (flagellar swimming and metaboly, see Leander et al., 2017) and capable of responding to a variety of stimuli, from light to confinement, see Noselli et al., 2019.
Materials and methods
Strain SAG 12245/27 of Euglena gracilis obtained from the SAG Culture Collection of Algae at the University of Göttingen was maintained axenic in liquid culture medium Eg. Cultures were transferred weekly. Cells were kept in an incubator at 15°C at a light:dark cycle of 12 hr under a cold white LED illumination with an irradiance of about $50\mu \text{mol}\cdot {\text{m}}^{2}\cdot {\text{s}}^{1}$.
An Olympus IX 81 inverted microscope with motorized stage was employed in all the experiments. These were performed at the Sensing and Moving Bioinspired Artifacts Laboratory of SISSA. The microscope was equipped with a LCAch 20X Phc objective (NA 0.40) for the imaging of cells trapped at the tip of a glass capillary using transmitted brightfield illumination. The intermediate magnification changer (1.6 X) of the microscope was exploited to achieve higher magnification. Micrographs were recorded at a frame rate of $1,000\text{fps}$ with a Photron FASTCAM Mini UX100 highspeed digital camera.
Tapered capillaries of circular cross section were obtained from borosilicate glass tubes by employing a micropipette puller and subsequently fire polished. At each trial, observation a glass capillary was filled with a diluted solution of cells and fixed to the microscope stage by means of a custom made, 3dprinted holder. The holder allowed for keeping the capillary in place and rotating it about its axis, so as to image a cell specimen from distinct viewpoints. Cells were immobilized at the tip of the capillary by applying a gentle suction pressure via a syringe connected to the capillary by plastic tubing. Occasionally, large body deformations of the cells were observed, a behavior commonly known as ‘metaboly’ which E. gracilis often manifest under confinement. The observations reported in this paper are restricted to specimens with immotile cell bodies, that is, in absence of metaboly. This choice allows for a clear capture of the flagellar beat. The absence of metaboly also suggests minimal impact on cell behavior in response to capillary entrapment.
Appendix 1
Model details
The Ax consists of a bundle of inextensible filaments of length $L$ (MTs) lying on a cylindrical surface of radius ${\rho}^{a}$. For simplicity, the model ignores the mechanical effects of radial spokes and the central pair. The axonemal surface is parametrized by the generalized cylindrical coordinates $z$ and $\varphi $ via the map
with ${\mathbf{\mathbf{r}}}^{a}$, ${\mathbf{\mathbf{d}}}_{1}$, and ${\mathbf{\mathbf{d}}}_{2}$ defined as in the main text. In Cicconofri et al., 2020, this surface is modelled as a continuous subject to active shear deformations. Here, we base our model on the explicit description of the individual MTs. Following Hilfinger and Jülicher, 2008, we suppose that the axonemal constraints confine MTs on the Ax surface at a fixed angular distance $\mathrm{\Delta}\varphi =2\pi /9$ between each other. More formally, for $j=1,\mathrm{\dots},9$, we define the centerline ${\mathbf{\mathbf{r}}}^{j}$ of the $j$th MT as ${\mathbf{\mathbf{r}}}^{j}(s)=\mathbf{\mathbf{C}}(s,{\varphi}_{j})$, where
The function $Z(s,\varphi )$ in Equation 29 is defined (implicitly) via the equality
From the definitions above follows $\parallel {\partial}_{s}{\mathbf{\mathbf{r}}}^{j}\parallel =1$, so that MTs are indeed inextensible and $s$ is their arc length. Moreover, the Taylor expansion of $\mathbf{\mathbf{C}}$ at the first order in ${\rho}^{a}$ gives the approximated formula (Equation 5), with ${\gamma}_{1}$ and ${\gamma}_{2}$ given by Equation 6.
We associate to the $j$th MT an orthonormal frame along ${\mathbf{\mathbf{r}}}^{j}$ given by the unit vectors
where $\mathbf{\mathbf{N}}=\mathrm{cos}\varphi {\mathbf{\mathbf{d}}}_{1}(Z)+\mathrm{sin}\varphi {\mathbf{\mathbf{d}}}_{2}(Z)$ is the (outer) unit normal to the cylindrical surface. The unit vectors (Equation 31) determine MTs’ crosssection orientations. The unit vectors ${\mathbf{\mathbf{e}}}_{1}^{j}(s)$ and ${\mathbf{\mathbf{e}}}_{2}^{j}(s)$ lie on the crosssection centered at ${\mathbf{\mathbf{r}}}^{j}(s)$, while ${\mathbf{\mathbf{e}}}_{3}^{j}(s)={\partial}_{s}{\mathbf{\mathbf{r}}}^{j}(s)$ is orthogonal to it. The (passive) elastic energy of the Ax is given by the sum of the MTs’ elastic energies
are the strains associated to the $j$th MT, while ${B}^{m}$ and ${C}^{m}$ are the MTs’ bending and twisting moduli (respectively). At the leading order approximation in ${\rho}^{a}$ we have
From Equation 33 and Equation 3 follows that ${\mathcal{W}}_{pas}^{a}$, as defined in Equation 32, can indeed be approximated by the right hand side of Equation 2, with ${B}^{a}=9{B}^{m}$ and ${C}^{a}=9{C}^{m}$.
We consider now MT sliding. Fixed a point ${\mathbf{\mathbf{r}}}^{j+1}(s)$ on the $(j+1)$th MT’s centerline, we look for the nearest point to ${\mathbf{\mathbf{r}}}^{j+1}(s)$ on the centerline ${\mathbf{\mathbf{r}}}^{j}$ of the $j$th MT. Such a point ${\mathbf{\mathbf{r}}}^{j}({s}^{*})$ (we can think of it as a projection) lies at some arc length ${s}^{*}$, which depends on $s$. We write ${s}^{*}={\mathrm{\Pi}}_{j}(s)$. We then define the sliding ${\sigma}_{j}(s)$ as the difference of the two arc lengths $s$ and ${\mathrm{\Pi}}_{j}(s)$. More formally,
The figure below illustrates the geometric idea behind Equation 34.
The active internal energy of the Ax is defined as minus the total mechanical work of the dyneins
where ${F}_{j}(s)$ are the sliding forces on the $j$th MT exerted by the dyneins on the $(j+1)$th MT, and ${\widehat{F}}_{j}$ are the singular sliding forces (on the $j$th MT exerted by the dyneins on the $(j+1)$th MT) concentrated at the distal end of the Ax. Taylor expanding (Equation 34) in ${\rho}^{a}$ we have, at the leading order,
with ${\gamma}_{1}$ and ${\gamma}_{2}$ given by Equation 6. From Equation 36 we have that ${\mathcal{W}}_{act}^{a}$, as defined in Equation 35, is approximated by the right hand side of Equation 4, with
In the remainder we give some details of the derivation of Equation 11 from Equation 7. Expanding Equation 10 at the leading order in ${\rho}^{p}$ gives
with ${\gamma}_{p\perp}(s)=\mathrm{sin}{\varphi}^{p}{\gamma}_{1}(s)+\mathrm{cos}{\varphi}^{p}{\gamma}_{2}(s)$. From Equation 8 and Equation 38, leading order calculations give ${V}_{1}\sim {\rho}^{p}$ and ${V}_{3}\sim {\rho}^{p}$, whereas ${V}_{2}\approx {\gamma}_{p\perp}/\sqrt{1+{\gamma}_{p\perp}^{2}}$. Linearizing in ${\gamma}_{p\perp}$ we have $V}_{2}={\gamma}_{p\mathrm{\perp}$, from which follows Equation 11.
Appendix 2
Dynamical equations
To derive Equations 17–20, it is convenient to introduce the quantities $M}_{1$, $M}_{2$ and $M}_{3$ defined via the following variational equality
These quantities can be interpreted as the local components of the flagellar moment
A direct calculation gives
where ${\gamma}_{p\perp}(s)=\mathrm{sin}{\varphi}^{p}{\gamma}_{1}(s)+\mathrm{cos}{\varphi}^{p}{\gamma}_{2}(s)$, as in the previous Section. We then write the variations $\delta {U}_{i}$ in terms of $\delta {\theta}_{i}$ (defined in the main text) obtaining
Combining Equations 39, 40 and Equation 42 we have
In the calculations above, we took variations with $\delta {\mathbf{r}}^{a}(0)=\delta \mathit{\theta}(0)=\mathbf{0}$, since we consider a flagellum with a clamped end at $s=0$. Then, the principle of virtual work (Equation 16) yields the following force and torque balance equations
Equations 17–20 are derived from Equation 43 and Equation 44, after some extra formal manipulations that we explain in the reminder.
We first introduce the local angular velocities
which are related to the strains via the following compatibility equations
Equation 46 follows from the identities ${\partial}_{s}{\partial}_{t}{\mathbf{\mathbf{d}}}_{i}={\partial}_{t}{\partial}_{s}{\mathbf{\mathbf{d}}}_{i}$, with $i=1,2,3$. We then rewrite the external forces and torques (Equation 15) in compact form as
Scalar multiplying the torque balance equation by ${\mathbf{\mathbf{d}}}_{1}$ and ${\mathbf{\mathbf{d}}}_{2}$ we obtain the expressions for the first two local components
of the Lagrange multiplier vector $\mathbf{\mathbf{\Lambda}}={\mathrm{\Lambda}}_{1}{\mathbf{\mathbf{d}}}_{1}+{\mathrm{\Lambda}}_{2}{\mathbf{\mathbf{d}}}_{2}+{\mathrm{\Lambda}}_{3}{\mathbf{\mathbf{d}}}_{3}$. Scalar multiplying the torque balance equation by ${\mathbf{\mathbf{d}}}_{3}$ gives
We rewrite the force balance equation as ${\partial}_{t}{\mathbf{\mathbf{r}}}^{a}=\mathbf{\mathbf{V}}{\left[{\mathbf{\mathbf{d}}}_{3}\right]}^{1}{\partial}_{s}\mathbf{\mathbf{\Lambda}}$ and, after differentiating both sides with respect to $s$ and then scalar multiplying by ${\mathbf{\mathbf{d}}}_{1}$, ${\mathbf{\mathbf{d}}}_{2}$, and ${\mathbf{\mathbf{d}}}_{3}$, we obtain
From Equation 41, 46, and Equation 49 we obtain Equation 18 by first differentiating with respect to $s$ both sides of Equation 49, and then by linearizing the resulting equation. Similarly, exploiting also Equation 48 this time, we differentiate with respect to $s$ and then linearize Equation 50 to obtain Equation 17. Boundary conditions in Equation 19 follow from Equation 44, whereas Equation 20 are derived from
which follows from the fixed end condition $\mathbf{r}}^{a}(0,t)=\mathbf{0$.
Given $U}_{1$, $U}_{2$, and $U}_{3$ the orthonormal frame ${\mathbf{\mathbf{d}}}_{1}$, ${\mathbf{\mathbf{d}}}_{2}$, and ${\mathbf{\mathbf{d}}}_{3}$ can be recovered by integrating the system of equations
which is derived from Equation 3. We can then recover the centerline ${\mathbf{\mathbf{r}}}^{a}$ by integrating ${\partial}_{s}{\mathbf{\mathbf{r}}}^{a}={\mathbf{\mathbf{d}}}_{3}$. The PFR centerline ${\mathbf{\mathbf{r}}}^{p}$ and the orthonormal frame ${\mathbf{\mathbf{g}}}_{1}$, ${\mathbf{\mathbf{g}}}_{2}$, and ${\mathbf{\mathbf{g}}}_{3}$ follow from Equation 9 and Equation 10.
Appendix 3
Numerical simulations I: equilibria
We define the nondimensional variables for arc length $x$, time $y$, strains $u}_{i$, shear forces $h}_{i$, and concentrated shear forces ${\widehat{h}}_{i}$ as follows
The equilibrium Equation 12, with Equation 22, is solved by seeking for a minimizer of the energy (Equation 23) in nondimensional form
In the formula above $\mathbf{\mathbf{u}}=({u}_{1},{u}_{2})$ is the nondimensional bending vector and ${u}_{i}^{*}={U}_{i}^{*}/{L}^{1}$, with $i=1,2$, are the nondimensional target bending strains. We find the minimizer using the gradient descent method
and $\alpha$ is a conveniently chosen step size. We initiate the algorithm with ${\mathbf{\mathbf{u}}}^{0}=({u}_{1}^{*},{u}_{2}^{*})$, and then we iterate it until $\parallel {\mathbf{\mathbf{u}}}^{n+1}{\mathbf{\mathbf{u}}}^{n}\parallel $ falls below a preset tolerance parameter.
In the figure above, we considered four different perturbations of a perfectly planar, Chlamydomonaslike, target bending strain, and calculated the resulting flagellar equilibrium configurations for different values of the parameter $\nu$. Each configuration presents a different (nonnull) torsional profile for $\nu =0$ (thick, light blue lines). For larger values of $\nu$, all perturbations lead to (roughly) similar configurations, with torsional peaks of alternate sign.
Appendix 4
Numerical simulations II: flagellar dynamics
We consider the nondimensional variables defined in Equation 54. The dynamic Equation 17 is recast, and then solved, in terms of the shear vector $\mathit{\bm{\gamma}}=({\gamma}_{1},{\gamma}_{2})$ where
After solving for $\mathit{\bm{\gamma}}$, we obtain the bending strains by differentiation. The equation for $\mathit{\bm{\gamma}}$ is
and $\mathbf{\mathbf{h}}=({h}_{1},{h}_{2})$. The corresponding boundary conditions are given by
where $\widehat{\mathbf{\mathbf{h}}}=({\widehat{h}}_{1},{\widehat{h}}_{2})$. The above (Equation 57) are pointwise conditions, which do not involve integral terms as in Equation 19. Avoiding this nonlocality allows for an easier numerical implementation. The finite difference scheme we employ to solve Equations 56–57 is illustrated in the remainder.
We consider the discrete time sequence ${y}_{n}=n\mathrm{\Delta}y$ with $n=0,1,2,\mathrm{\dots}$ and we define ${\mathit{\bm{\gamma}}}^{n}(x)=\mathit{\bm{\gamma}}({y}_{n},x)$, ${\mathbf{\mathbf{h}}}^{n}(x)=\mathbf{\mathbf{h}}({y}_{n},x)$, and ${\widehat{\mathbf{\mathbf{h}}}}^{n}=\widehat{\mathbf{\mathbf{h}}}({y}_{n})$. Equation 56 is discretized in time with the onestep (semiimplicit) numerical scheme
and complemented by the boundary conditions
The nondimensional arc length interval $[0,1]$ is discretized uniformly in $M+1$ points ${x}_{k}=k\mathrm{\Delta}x$ , with $k=0,1,\mathrm{\dots},M=1/\mathrm{\Delta}x$. We also consider the extra 'ghost points’ ${x}_{k}=k\mathrm{\Delta}x$ with $k=1,M+1,M+2$. The discrete values ${\mathrm{\partial}}_{x}^{2}{\mathit{\gamma}}_{k}^{n}={\mathrm{\partial}}_{x}^{2}{\mathit{\gamma}}^{n}({x}_{k})$ of the second derivative in Equation 56 are approximated by the finite difference scheme
where ${\mathit{\bm{\gamma}}}_{k}^{n}={\mathit{\bm{\gamma}}}^{n}({x}_{k})$. Analogous formulas are employed for the second derivative of ${\mathbf{\mathbf{h}}}^{n}$. The discrete values ${\partial}_{x}^{4}{\mathit{\bm{\gamma}}}_{k}^{n+1}={\partial}_{x}^{4}{\mathit{\bm{\gamma}}}^{n+1}({x}_{k})$ of the forth derivative in Equation 56 are given by the scheme
which involves the ghost points values ${\mathit{\bm{\gamma}}}_{1}$, ${\mathit{\bm{\gamma}}}_{M+1}$, and ${\mathit{\bm{\gamma}}}_{M+2}$. The discretized approximations of the boundary conditions (Equation 59) are given by
The previous formulas give us the expressions for the ghost points’ values ${\mathit{\bm{\gamma}}}_{1}^{n+1}$, ${\mathit{\bm{\gamma}}}_{M+1}^{n+1}$, and ${\mathit{\bm{\gamma}}}_{M+2}^{n+1}$ in terms of ${\widehat{\mathbf{\mathbf{h}}}}^{n+1}$, ${\mathit{\bm{\gamma}}}_{k}^{n+1}$, ${\mathit{\bm{\gamma}}}_{k}^{n}$, and ${\mathbf{\mathbf{h}}}_{k}^{n}$ with $0\le k\le M$. These expressions are then plugged in Equation 61. In turn, the iterative scheme (Equation 58) allows to calculate ${\mathit{\bm{\gamma}}}_{k}^{n+1}$ from ${\mathit{\bm{\gamma}}}_{k}^{n}$ with $0\le k\le M$, while incorporating the boundary conditions (Equation 59) in the numerical solution. The scheme is iterated for several time periods until a periodic solution is reached.
We obtain the history of shear forces presented in Figure 7 by solving the following inverse dynamical problem. We assign first a history of normalized bending strains $(0,{u}_{2}(x,y))$, periodic in time, that imitates the experimentally observed Chlamydomonas flagellar curvatures reported in Qin et al., 2015. We use the following model
Then, we calculate the shear forces that generate said history of bending strains for an Ax beating in a viscous fluid (without extraaxonemal structures attached to it). Equation 56 with $\nu =0$ and ${h}_{2}={\widehat{h}}_{2}=0$ defines exactly the dynamics of this system. We can solve for $h}_{1$ and ${\widehat{h}}_{1}$ explicitly, obtaining ${\widehat{h}}_{1}(y)={u}_{2}(1,y)$ and ${h}_{1}(x,y)={\partial}_{x}{u}_{2}(x,y)\eta {\int}_{x}^{1}{\int}_{0}^{{x}^{\prime}}{\int}_{0}^{{x}^{\prime \prime}}{\partial}_{y}{u}_{2}({x}^{\prime \prime \prime},y)\mathit{d}{x}^{\prime \prime \prime}\mathit{d}{x}^{\prime \prime}\mathit{d}{x}^{\prime}.$
In the dynamic simulation shown in Figure 7, we used the following numeric values for the physical parameters of the system. The bending modulus of the Ax is $B}^{a}=840\phantom{\rule{mediummathspace}{0ex}}\text{pN}\cdot \mu {\text{m}}^{2$, taken from Xu et al., 2016. We set $L=28\phantom{\rule{mediummathspace}{0ex}}\mu \text{m}$, $T=25\phantom{\rule{mediummathspace}{0ex}}\text{ms}$, and $\mu}_{\mathrm{\perp}}=3.1\phantom{\rule{mediummathspace}{0ex}}\text{fN}\cdot s\cdot \mu {\text{m}}^{2$, which are all values estimated in Rossi et al., 2017. The angle between spontaneous bending plane and the AxPFR joining line ${\varphi}^{p}=2\pi /9$ is estimated from micrographs in Melkonian et al., 1982 and Bouck et al., 1990. For the bending strains parameters in Equation 62 we took ${a}_{0}=7.8$, ${a}_{1}=7.5$, ${\lambda}_{0}=1.85$, ${\lambda}_{1}=0.1$, and ${\omega}_{1}=0.1$.
Without direct measurements for ${D}^{p}$, we set $\nu =20$ as the value that best replicates the shapes and tridimensionality of the Euglena beat. The nondimensional parameter $\nu ={D}^{p}/{B}^{a}{L}^{2}$ can be defined as the ratio between the elastic energy of an Ax bent with constant curvature $1/L$ and the elastic energy of the PFR deformed with uniform shear $\gamma =1$. The two deformations do not happen simultaneously during flagellar deformation, where the kinematics of Ax and PFR are coupled. For this reason, the numerical value of parameter $\nu$ does not give a reliable estimate of the relative contribution ox Ax and PFR to the total elastic energy during flagellar deformation. One can define an effective parameter ${\nu}_{e}$ to measure the interplay of PFR and Ax elasticity in the following way. We can consider the elastic energy of an Ax bent along the spontaneous bending plane with constant curvature $1/L$, thus ${U}_{1}=0$ and ${U}_{2}=1/L$. This deformation of the Ax induces a PFR shear deformation with ${\gamma}_{1}=s/L$ and ${\gamma}_{2}=0$ (due to the kinematic constraints, see ‘Mechanical model’ Section). In this case, the (passive) elastic energy of the Ax is given by ${\mathcal{W}}_{pas}^{a}=\frac{1}{2}{B}^{a}{L}^{1}$, while the elastic energy of the PFR is given by ${\mathcal{W}}^{p}=\frac{1}{6}L{D}^{p}{\mathrm{sin}}^{2}({\varphi}^{p})$. We define the effective parameter ${\nu}_{e}$ as the ratio between these two energies, which gives ${\nu}_{e}=\frac{1}{3}{\mathrm{sin}}^{2}({\varphi}^{p})\nu $. For ${\varphi}^{p}=2\pi /9$, the value for the angle we used for E. gracilis, we have ${\nu}_{e}=0.13\nu $. Thus, for $\nu =20$ we have ${\nu}_{e}=2.75$. This value of ${\nu}_{e}$ is close to one, showing that our simulations lead to large nonplanarity of the flagellar beat when the two competing elastic energies of Ax and PFR are of similar magnitude. This is in agreement with our hypothesis that nonplanarity emerges form competing and frustrated elastic structures, via the principle of structural incompatibility.
Appendix 5
Numerical simulations III: swimming
We associate to the (elliptical) cell body a moving frame of orthonormal unit vectors $\mathbf{\mathbf{i}}$, $\mathbf{\mathbf{j}}$ and $\mathbf{\mathbf{k}}$, aligned with the major axis of the cell, see the figure below. The moving frame is located at the geometric center of the body $\mathbf{\mathbf{c}}$. We denote with $\overline{\mathbf{\mathbf{r}}}(s,t)$ the flagellum centerline in the moving frame coordinates. In the lab frame coordinates, the centerline of the flagellum is thus given by
where $\mathbf{\mathbf{Q}}(t)$ is the cell body rotation, which has the columnwise matrix expression
In our simulation, the (periodic) time history of curves $\overline{\mathbf{\mathbf{r}}}(s,t)$ is given by the the flagellar shapes obtained from our model, shown in the ‘Hydrodynamic simulations’ Section of the main text. The body center trajectory $\mathbf{\mathbf{c}}(t)$ and the rotation $\mathbf{\mathbf{Q}}(t)$ are quantities to be obtained from the swimming simulation. We have
where $\mathit{\bm{\omega}}(t)$ is the rotational velocity of the cell body, $\overline{\mathbf{\mathbf{v}}}={\mathbf{\mathbf{Q}}}^{T}\dot{\mathbf{\mathbf{c}}}$, and $\overline{\mathit{\bm{\omega}}}={\mathbf{\mathbf{Q}}}^{T}\mathit{\bm{\omega}}$. From the force balance equations of the swimmer we can derive a formula for the quantities $\overline{\mathbf{\mathbf{v}}}$ and $\overline{\mathit{\bm{\omega}}}$, which are completely determined by $\overline{\mathbf{\mathbf{r}}}$ and the geometry of the cell body (which are given quantities). The hydrodynamic forces acting on the flagellum and the cell body are approximated using Resistive Force Theory (Equation 15). The linear density force $\mathbf{\mathbf{f}}(s,t)$ at time $t$ exerted by the surrounding fluid can be written as
where ${\mathbf{\mathbf{v}}}_{}={\partial}_{s}\mathbf{\mathbf{r}}({\partial}_{s}\mathbf{\mathbf{r}}\cdot {\partial}_{t}\mathbf{\mathbf{r}})$ and ${\mathbf{\mathbf{v}}}_{\perp}={\partial}_{t}\mathbf{\mathbf{r}}{\mathbf{\mathbf{v}}}_{}$ are the local velocity components parallel and perpendicular to the flagellum. We have
The total viscous force ${\mathbf{\mathbf{f}}}_{tot}$ and torque ${\mathbf{\mathbf{g}}}_{tot}$ acting on the swimmer are given by
where ${\mathbf{\mathbf{f}}}_{flag}$ and ${\mathbf{\mathbf{g}}}_{flag}$ are total viscous force and torque acting on the flagellum
while ${\mathbf{\mathbf{f}}}_{body}$ and ${\mathbf{\mathbf{g}}}_{body}$ are the total force and torque acting on the cell body
For the detailed expressions of the viscous resistive coefficients ${A}_{}$, ${A}_{\perp}$, ${D}_{}$, and ${D}_{\perp}$ see, for example Kim and Karrila, 2013. The force and torque balance on the cell gives us the equations of swimming dynamics
which lead to the following formula
where
In the identities above ${\left[\mathbf{\mathbf{a}}\right]}_{\times}$ denotes the skewsymmetric matrix generated by the vector $\mathbf{\mathbf{a}}$, which is defined as the matrix such that ${\left[\mathbf{\mathbf{a}}\right]}_{\times}\mathbf{\mathbf{b}}=\mathbf{\mathbf{a}}\times \mathbf{\mathbf{b}}$ for every vector $\mathbf{\mathbf{b}}$. The time evolution of $\mathbf{\mathbf{c}}(t)$ and $\mathbf{\mathbf{Q}}(t)$ are then obtained by integrating the equations
Results of the simulation are shown in Figure 8 and in Video 3. In the figure below we show the calculated components of the vectors $\overline{\mathbf{\mathbf{v}}}=(\mathbf{\mathbf{v}}\cdot \mathbf{\mathbf{i}},\mathbf{\mathbf{v}}\cdot \mathbf{\mathbf{j}},\mathbf{\mathbf{v}}\cdot \mathbf{\mathbf{k}})$ and $\overline{\mathit{\bm{\omega}}}=(\mathit{\bm{\omega}}\cdot \mathbf{\mathbf{i}},\mathit{\bm{\omega}}\cdot \mathbf{\mathbf{j}},\mathit{\bm{\omega}}\cdot \mathbf{\mathbf{k}})$ during a flagellar beat. We considered an ellipsoidal cell $42\mu \text{m}$ long and $7.2\mu \text{m}$ wide, with a flagellum of length $L=28\mu \text{m}$. The flagellum shapes are those illustrated in Figure 7, and we use for the resistive coefficients the values estimated in Rossi et al., 2017.
The simulations show a good qualitative agreement with the experimentally observed trajectories. They predict a (generalized) helical trajectory with radius $\approx 1.4\mu \text{m}$ and with the cell closing a full turn of the helix in $\sim 24$ beats, in excellent agreement with observations of E. gracilis specimens (Rossi et al., 2017). The pitch of the numerical trajectory is $\approx 9.5\mu \text{m}$, $\sim 0.2$ times the length of the cell which is less then typically observed values of $\sim 0.7$.

Appendix 5—figure 1—source code 1
 https://cdn.elifesciences.org/articles/58610/elife58610app5fig1code1v2.zip
Data availability
All data generated or analysed during this study are included in the manuscript and supporting files. Source data and code files have been provided for Figures 2, 6, and 7 in the main text, and for Figure 1 in Appendix 5.
References

BookMolecular Biology of the CellIn: Richter M. L, editors. Molecular Sciences. New York: Garland Science. pp. 941–942.

Simulating the complex cell design of Trypanosoma brucei and its motilityPLOS Computational Biology 11:e1003967.https://doi.org/10.1371/journal.pcbi.1003967

BookEuglena gracilis a model for flagellar surface assembly, with reference to other cells that bear flagellar mastigonemes and scalesIn: Bloodgood R. A, editors. Ciliary and Flagellar Membranes. Springer. pp. 65–90.https://doi.org/10.1007/9781461305156_3

BookMovement and locomotion of euglenaIn: Rothstein M, editors. The Biology of Euglena. Vol. III: Physiology. Academic Press. pp. 143–168.https://doi.org/10.1007/BF01279733

Thinking about flagellar oscillationCell Motility and the Cytoskeleton 66:425–436.https://doi.org/10.1002/cm.20313

The paraflagellar rod: a structure in search of a functionBiology of the Cell 63:169–181.https://doi.org/10.1016/02484900(88)900561

Morphable structures from unicellular organisms with active, shapeshifting envelopes: variations on a theme by gaussInternational Journal of NonLinear Mechanics 118:103278.https://doi.org/10.1016/j.ijnonlinmec.2019.103278

Antagonistic behaviors of NMY1 and NMY2 maintain ring channels in the C. elegans GonadBiophysical Journal 111:2202–2213.https://doi.org/10.1016/j.bpj.2016.10.011

The elastic basis for the shape of Borrelia burgdorferiBiophysical Journal 96:4409–4417.https://doi.org/10.1016/j.bpj.2009.02.066

Flagellar and ciliary beating in trypanosome motilityCell Motility and the Cytoskeleton 64:629–643.https://doi.org/10.1002/cm.20210

How Euglena gracilis swims: flow field reconstruction and analysisPhysical Review. E 103:023102.https://doi.org/10.1103/PhysRevE.103.023102

Flagellum assembly and function during the leishmania life cycleCurrent Opinion in Microbiology 13:473–479.https://doi.org/10.1016/j.mib.2010.05.008

Green algae as model organisms for biological fluid dynamicsAnnual Review of Fluid Mechanics 47:343–375.https://doi.org/10.1146/annurevfluid010313141426

BookThe Mathematics and Mechanics of Biological GrowthNew York: SpringerVerlag.https://doi.org/10.1007/9780387877105

The chirality of ciliary beatsPhysical Biology 5:016003.https://doi.org/10.1088/14783975/5/1/016003

The diversity of hydrostatic skeletonsJournal of Experimental Biology 215:1247–1257.https://doi.org/10.1242/jeb.056549

BookPrinciples and selected applicationsIn: Nitsche L. C, editors. Microhydrodynamics. Courier Corporation. pp. 739–743.https://doi.org/10.1002/aic.690400418

Motility and more: the flagellum of Trypanosoma bruceiNature Reviews Microbiology 12:505–518.https://doi.org/10.1038/nrmicro3274

BookHandbook of the ProtistsSpringer International Publishing.https://doi.org/10.1007/9783319326696

Flagellar and ciliary beating: the proven and the possibleJournal of Cell Science 123:519–528.https://doi.org/10.1242/jcs.051326

Flagellar membrane specializations and their relationship to mastigonemes and microtubules in Euglena gracilisJournal of Cell Science 55:115–135.

Anisotropic ESCRTIII architecture governs helical membrane tube formationNature Communications 11:1516.https://doi.org/10.1038/s41467020153274

ATPase activity in flagella from Euglena gracilis localization of the enzyme and effects of detergentsThe Journal of Protozoology 22:331–335.https://doi.org/10.1111/j.15507408.1975.tb05182.x

The paraflagellar rod of kinetoplastid parasites: from structure to components and functionInternational Journal for Parasitology 40:135–148.https://doi.org/10.1016/j.ijpara.2009.10.005

Ultrastructure of the apical zone of Euglena gracilis: photoreceptors and motor apparatusElectron Microscopy Reviews 4:319–342.https://doi.org/10.1016/08920354(91)90008Z

Swimming like algae: biomimetic soft artificial ciliaJournal of the Royal Society Interface 10:20120666.https://doi.org/10.1098/rsif.2012.0666

Curvature regulation of the ciliary beat through axonemal twistPhysical Review E 94:042426.https://doi.org/10.1103/PhysRevE.94.042426

Coronavirus envelope protein: current knowledgeVirology Journal 16:69.https://doi.org/10.1186/s1298501911820

A comparison of paraxial rods in the flagella of euglenoids and kinetoplastidsArchiv Für Protistenkunde 143:177–194.https://doi.org/10.1016/S00039365(11)802873

On the unity and diversity of ciliaPhilosophical Transactions of the Royal Society B: Biological Sciences 375:20190148.https://doi.org/10.1098/rstb.2019.0148

Use of chiral cell shape to ensure highly directional swimming in trypanosomesPLOS Computational Biology 13:e1005353.https://doi.org/10.1371/journal.pcbi.1005353

Twirling and whirling: viscous dynamics of rotating elastic filamentsPhysical Review Letters 84:1623–1626.https://doi.org/10.1103/PhysRevLett.84.1623

Evidence for a slidingresistance at the tip of the trypanosome flagellumCell Motility and the Cytoskeleton 63:741–746.https://doi.org/10.1002/cm.20159
Decision letter

Raymond E GoldsteinReviewing Editor; University of Cambridge, United Kingdom

Aleksandra M WalczakSenior Editor; École Normale Supérieure, France
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.
Acceptance summary:
The large photosynthetic biflagellate Euglena displays an intriguing swimming dynamics. The single emergent flagellum is held in front of the cell, where it executes a highly3D, socalled spinning lasso beat pattern. Unusually, the euglena flagellum is flanked by a paraflagellar or paraaxial rod structure, whose function is so far unknown. The authors put forward an interesting proposition (based on geometric mechanics) that a structural incompatibility between the two filaments leads to frustration and a torsional signature, consistent with the lasso. Both the static case and the hydrodynamic case were tested – the torsional feature seems to be robust to external forcing.
Decision letter after peer review:
Thank you for submitting your article "The biomechanical role of extraaxonemal structures in shaping the flagellar beat of Euglena" for consideration by eLife. Your article has been reviewed by three peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by Aleksandra Walczak as the Senior Editor. The reviewers have opted to remain anonymous.
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
We would like to draw your attention to changes in our revision policy that we have made in response to COVID19 (https://elifesciences.org/articles/57162). Specifically, we are asking editors to accept without delay manuscripts, like yours, that they judge can stand as eLife papers without additional data, even if they feel that they would make the manuscript stronger. Thus the revisions requested below only address clarity and presentation.
Summary:
The large photosynthetic biflagellate Euglena displays an intriguing swimming dynamics. The single emergent flagellum is held in front of the cell, where it executes a highly3D, socalled spinning lasso beat pattern. Unusually, the euglena flagellum is flanked by a paraflagellar or paraaxial rod structure, whose function is so far unknown. The authors put forward an interesting proposition (based on geometric mechanics) that a structural incompatibility between the two filaments leads to frustration and a torsional signature, consistent with the lasso. Both the static case and the hydrodynamic case were tested – the torsional feature seems to be robust to external forcing.
The reviewers found this to be a clearly written and wellexplained paper that should be of interest to a wide range of researchers in various fields of biology and biophysics. While the level of mathematics is nontrivial, it is explained well within the text, and illustrated well in the figures.
Essential revisions:
1) The numbering of MTs is chosen by the authors does not appear to be consistent with published literature. This is important and worth checking carefully, in order to crossreference any previous structural or EM studies. Usually, authors adopt a certain convention when it comes to numbering of microtubule doublets, as such systems do not exhibit perfect rotational symmetry but retain many heterogeneities – and there could be distinguished associations between certain Mts and other, nonaxonemal structures such as mastigonemes.
2) According to the literature, e.g. Hyams, 1982, the PFR is hollow (Figure 3 of that paper). In contrast in T. brucei there are some longitudinal filaments and more like a lattice structure. Does this matter? The ventral flagellum also has a PFR – does this serve any purpose? Also, in euglena the rod extends all the way along the flagellum, but this is not the case in some other species. Do the authors believe that their conclusion extends across phyla, and if not, why not?
a) There is earlier work on the shape of spirochetes where a similar issue of two elastically competing structures appears: Dombrowski, et al., (2009).
The authors may wish to comment on the similarities and differences between that study and theirs.
3) In many other types of cilia and flagella which exhibit strongly nonplanar beats, there is no association with a PFRlike structure, i.e. the latter is not necessary to generate nonplanar bending. What's different here? Would be good to make and discuss some more crossspecies comparisons (e.g. see Koyfman et al., 2011)
4) Is it clear that the rod assumes a consistent relationship with the numbered doublets? Could the rod itself be twisted around the axoneme? Authors assumed a fixed relationship and phi_p angle, and that this is neither perpendicular nor parallel to the beat plane… this is important according to Equations 25/26 (make sure to cite Melkonian, 1982 when mentioning the phi_p angle.)
5) In order to test their hypothesis the authors implemented a Chlamydomonas like planar beat pattern for dynein actuation, and determined the mismatch between the rod and axoneme… however, why could the naked axonemal beat not be 3D in the first place? Is there any structural evidence (as exists in Chlamydomonas) which suggests that the euglena beat is planar?
6) The authors have also assumed that the PFR, together with the AX_PFR linkages are passive. We suspect this is not the case. The fact that the PFR can be digested easily by trypsin suggests it may be contractile. The question is then could this structure participate in phototaxis – what is known about how the beat pattern changes during phototaxis – could the PFR actively enable a faster change in beat pattern or even beat frequency – via active contraction say, upon perception of photic signals? This could potentially be testable in the model.
7) Finally, the paraaxial rod could also provide or promote a number of other functions, for instance gliding motility. What do the authors think about this possibility? Or conversely, can they suggest experiments that would help show more definitively that the PFR facilities 3D swimming?
8) The authors state (subsection “Mechanical model') that "Classical estimations on homogeneous elastic rods, see, e.g., Goriely, (2017), show that bending and twist moduli scale with the forth power of the cross section radius, whereas shear and stretching moduli scale with the second power and hence they are dominant for small radii." That there is a crossover is clear, but numbers matter here: For realistic material properties what is the length scale of the crossover?
(Also, "forth">"fourth").
9) Although the mechanical description is general, the main claim of the paper that a sequence of shapes with the observed characteristics can emerge from a combination of a passive and active component with the given constrains is supported.
It would be interesting to know if the application of a hydrodynamic model to the theoretical waveforms can produce swimming paths similar to the experimentally observed ones.
10) The conclusion of the paper is of general nature, thus it could be noted that even in cilia with quasiplanar beats (like is the case for Chlamydomonas) asymmetric sliding constrains exist and that it is still debated if those are involved in generating axonemal twist or torsional waves during the beat. Therefore, it could be interesting to also discuss the effect of a weakly constrained beat 'weak paraflagella rod'.
How does the spinninglasso beat breaks down?
What signature would waveforms under those conditions have?
https://doi.org/10.7554/eLife.58610.sa1Author response
Essential revisions:
1) The numbering of MTs is chosen by the authors does not appear to be consistent with published literature. This is important and worth checking carefully, in order to crossreference any previous structural or EM studies. Usually, authors adopt a certain convention when it comes to numbering of microtubule doublets, as such systems do not exhibit perfect rotational symmetry but retain many heterogeneities – and there could be distinguished associations between certain Mts and other, nonaxonemal structures such as mastigonemes.
We have changed the numbering of MTs to make it consistent with published literature on E. gracilis EM studies. Specifically, we adopted the numbering convention of Melkonian et al., (1982) and Bouck et al., (1990). MT 1 is defined as the doublet with the most prominent attachment to the PFR. In this convention, MTs’ numbering increases in the clockwise direction when seen from the distal end of the Ax. This is at odds with the MTs’ labelling typically used in structural studies of cilia (which is the one we adopted in the first submission of our paper). In the typical case, MTs’ numbering increases in the anticlockwise direction when seen from the distal end of the Ax. Dynein arms extend from MT N and reach MT N+1. In the convention adopted in this revised version of our manuscript, dynein arms extend from MT N and reach MT N1, as in Figure 14 in Melkonian et al., (1982) and Figure 2 in Bouck et al., (1990).
2) According to the literature, e.g. Hyams, 1982, the PFR is hollow (Figure 3 of that paper). In contrast in T. brucei there are some longitudinal filaments and more like a lattice structure. Does this matter? The ventral flagellum also has a PFR – does this serve any purpose? Also, in euglena the rod extends all the way along the flagellum, but this is not the case in some other species. Do the authors believe that their conclusion extends across phyla, and if not, why not?
Our model for the PFR as a homogeneous elastic rod does not capture the level of detail that the reviewer is aiming for. We consider it as an “effective' coarsegrained model giving a fair first approximation of the forces of elastic origin that the PFR is subject to and transmits to the Ax. Through a suitable choice of geometric and material parameters, it can be used to model both hollow structures and structures with a dense core. Therefore, we believe that our conclusions are robust even against uncertainty of *some* structural details. See answer to no. 3 for more comments on this issue (in particular, comparisons across species).
The question about the ventral flagellum is interesting but beyond the scope of our work.
a) There is earlier work on the shape of spirochetes where a similar issue of two elastically competing structures appears: Dombrowski, et al., (2009).
The authors may wish to comment on the similarities and differences between that study and theirs.
We thank the reviewers for pointing out the study by Dombrowski et al., to us. We have added it to our reference list, and commented on the similarities between the mechanism of shapemorphing in B. burgdorferi and that of flagellar beating in E. gracilis. Both mechanisms rely on mechanical “competition' between substructures, which generates frustration resulting in nontrivial shapes.
3) In many other types of cilia and flagella which exhibit strongly nonplanar beats, there is no association with a PFRlike structure, i.e. the latter is not necessary to generate nonplanar bending. What's different here? Would be good to make and discuss some more crossspecies comparisons (e.g. see Koyfman et al., 2011)
Yes, the association with a PFRlike structure is not necessary to generate nonplanarity in flagella. We did not imply otherwise, but we have made this point clearer in our resubmission. Our main conclusion applies to E. gracilis specifically, where a prominent asymmetry in the PFRAx bonding complex is present. In our model, the PFRAx interaction is needed to generate the typical spinning lasso geometrical signature which defines the peculiar Euglena beating style.
In other PFRbearing flagella, like those of Leishmania mexicana and Trypanosoma brucei, such a prominent asymmetry is lacking, see e.g. Portman and Gull, (2010). For these organisms, the angle ${\varphi}^{p}$ is closer to zero, as we mentioned in our first submission. In this case, our model predicts a planar (or quasi planar) beat and this prediction is consistent with the observed behaviour of both L. mexicana and T. brucei mutants with bodydetached flagellum. Following the suggestion by the reviewer, we have added a reference to Koyfman et al., (2011).
4) Is it clear that the rod assumes a consistent relationship with the numbered doublets? Could the rod itself be twisted around the axoneme? Authors assumed a fixed relationship and phi_p angle, and that this is neither perpendicular nor parallel to the beat plane… this is important according to Equations 25/26 (make sure to cite Melkonian, 1982 when mentioning the phi_p angle.)
We believe that the EM study by Melkonian et al., (1982) gives a clear answer to the question raised by the reviewer. The possibility of a systematic numbering convention of MTs strongly implies a consistent relationship between Ax and PFR. The authors define a scheme for the flagellar crosssection “based on the evaluation of 80 different crosssections of flagella', where MTs are identified based on associations with PFR and mastigonemes. The bonding links complex is conserved through the length of the flagellum (and across flagella), and it seems hard to reconcile this observation with the possibility that the PFR was twisted around the Ax. We cited Melkonian et al., (1982) in the paper when mentioning the ${\varphi}^{p}$ angle.
5) In order to test their hypothesis the authors implemented a Chlamydomonas like planar beat pattern for dynein actuation, and determined the mismatch between the rod and axoneme… however, why could the naked axonemal beat not be 3D in the first place? Is there any structural evidence (as exists in Chlamydomonas) which suggests that the euglena beat is planar?
In our model the role of the PFR is twofold: it acts as (1) a MTs’ sliding inhibitor, and (2) as a mechanically antagonistic element.
The AxPFR bonding links play the central (structural) role in MTs’ sliding inhibition. Our working hypothesis is that MTs’ sliding stretches the bonding links. This inhibits MTs’ sliding and triggers a dynein organization (via mechanical feedback) similar to the one present in Chlamydomonas and sperm cells. However, we are not yet able to model the microscopic details of this response mechanism. We take this feedbackbased selforganization process as a given, and we consider a force pattern that would produce planar beat in absence of antagonistic forces on the Ax.
The agreement of the observed flagellar shapes with the ones we predict shows that our working hypothesis is consistent with the available evidence. Further work will be needed in the future to address the issue of whether the dynein actuation pattern is truly planar and, if so, which feedback mechanisms lead to the emergence of this pattern. We have clarified this point in the revised manuscript (see subsection “Dyneins’ actuation induced by sliding inhibition').
6) The authors have also assumed that the PFR, together with the AX_PFR linkages are passive. We suspect this is not the case. The fact that the PFR can be digested easily by trypsin suggests it may be contractile. The question is then could this structure participate in phototaxis – what is known about how the beat pattern changes during phototaxis – could the PFR actively enable a faster change in beat pattern or even beat frequency – via active contraction say, upon perception of photic signals? This could potentially be testable in the model.
This is an interesting and widely open question, but we believe that the answer is highly nontrivial. The question deserves further adhoc study, including a large set of new experiments and a modelling effort to address the mechanisms of active contraction, light perception, and their interplay. We consider this suggestion as a very interesting direction to pursue in future studies, and we have indicated so in Discussion section.
7) Finally, the paraaxial rod could also provide or promote a number of other functions, for instance gliding motility. What do the authors think about this possibility? Or conversely, can they suggest experiments that would help show more definitively that the PFR facilities 3D swimming?
This is another interesting question, but we are not able to suggest an answer.
In other systems, the question of whether the paraflagellar rod is “vital' for a certain form of motility has been address with the study of mutants (see e.g., Bastin et al., 1998).Genetic editing techniques are not available to us, and we are unaware of results of this nature in the general literature on E. gracilis (possibly because its genome has not been sequenced yet).
Our experience with physical experiments on E. gracilis concerns swimming motility and metaboly, and we have no experience with gliding motility. We have not been able (at least not yet) to device a single physical experiment proving or disproving that PFR facilitates 3D swimming.
8) The authors state (subsection “Mechanical model') that "Classical estimations on homogeneous elastic rods, see, e.g., Goriely, (2017), show that bending and twist moduli scale with the forth power of the cross section radius, whereas shear and stretching moduli scale with the second power and hence they are dominant for small radii." That there is a crossover is clear, but numbers matter here: For realistic material properties what is the length scale of the crossover?
Consider the bending energy ${W}_{b}$ and the shear energy ${W}_{s}$ of a deformed homogeneous rod with circular crosssections of radius $\rho $. Suppose that external forces induce a bending of the rod with curvature $1/L$, and a shear $\gamma $. The bending and shear energies are given by ${W}_{b}=\frac{1}{2}B{L}^{1}$ and ${W}_{s}=\frac{1}{2}\mathrm{\text{LD}}{\gamma}^{2}$, respectively. The bending stiffness $B$ and the shear stiffness $D$ are given by
$B=\frac{\mathrm{\text{Y\pi}}{\rho}^{4}}{4}\phantom{\rule{0.222em}{0ex}}$ and $D=\frac{\mathrm{\text{Y \pi}}{\rho}^{2}}{2(1+p)\chi}\phantom{\rule{0.222em}{0ex}},$
where $\phantom{\rule{0.333em}{0ex}}\mathrm{\text{Y}}$ is the Young modulus of the rod, $\phantom{\rule{0.333em}{0ex}}\mathrm{\text{p}}$ is the Poisson ratio, and $\chi =\frac{32}{27}$ is a geometric coefficient. The bending energy is negligible with respect to the shear energy if $\frac{{W}_{s}}{{W}_{b}}\gg 1$. For the PFR of E. gracilis we have $L\sim 25\phantom{\rule{0.222em}{0ex}}\mu m$ and $\rho \sim 80\phantom{\rule{0.222em}{0ex}}nm$, see, e.g., Walne and Dawson, (1993). Thus, for every value of the Young modulus, and for a Poisson ratio within the typical range of 0 and ½, we have $\frac{{W}_{s}}{{W}_{b}}\sim {10}^{5}{\gamma}^{2}$. Even for exceedingly small shear angles $\gamma \sim 0.03$ the strain energy dominates over the bending energy by two orders of magnitude $\frac{{W}_{s}}{{W}_{b}}\sim {10}^{2}.$ For larger (and more realistic) shear angles this dominance increases.
(Also, "forth">"fourth").
We have corrected the typo.
9) Although the mechanical description is general, the main claim of the paper that a sequence of shapes with the observed characteristics can emerge from a combination of a passive and active component with the given constrains is supported.
It would be interesting to know if the application of a hydrodynamic model to the theoretical waveforms can produce swimming paths similar to the experimentally observed ones.
We have followed this suggestion. We have shown that the sequence of flagellar shapes obtained from our model (shown in subsection “Hydrodynamic simulations and comparison with observations', Figure 7) can generate swimming paths similar to those observed in experiments. Swimming cell simulation results are reported in Figure 8 in the main text, Appendix 5, and in Video 3. The results are briefly reviewed in the Discussion section, and more details on the implementation and physics behind swimming simulations are given in Appendix 5.
10) The conclusion of the paper is of general nature, thus it could be noted that even in cilia with quasiplanar beats (like is the case for Chlamydomonas) asymmetric sliding constrains exist and that it is still debated if those are involved in generating axonemal twist or torsional waves during the beat. Therefore, it could be interesting to also discuss the effect of a weakly constrained beat 'weak paraflagella rod'.
How does the spinninglasso beat breaks down?
What signature would waveforms under those conditions have?
We have performed a sensitivity analysis of flagellar shapes with spontaneous Chlamydomonaslike bending under different perturbations, see Appendix 3. We relaxed the planar constraint and considered four different weakly nonplanar spontaneous configurations. Each configuration presents a different (nonnull) torsional profile when the antagonistic mechanical interaction of the PFR is absent $\nu =0$. For larger values of $\nu $, each perturbation assumes the typical spinning lasso geometry (torsional peaks of alternate sign), regardless of the native ($\nu =0$) torsional profile. This shows that the PFRAx interaction strongly influence the shape outcome even when the perfectly planar constraint on the spontaneous bending is relaxed.
https://doi.org/10.7554/eLife.58610.sa2Article and author information
Author details
Funding
European Research Council (340685MicroMotility)
 Antonio DeSimone
Ministero dell’Istruzione, dell’Università e della Ricerca (Dipartimenti di Eccellenza 20182022 (SISSA  Area Matematica; Scuola Superiore Sant'Anna  Department of Excellence in Robotics and AI))
 Giovanni Noselli
 Antonio DeSimone
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank A Beran for his assistance with E. gracilis samples. This study was supported by the European Research Council through the ERC Advanced Grant 340685MicroMotility.
Senior Editor
 Aleksandra M Walczak, École Normale Supérieure, France
Reviewing Editor
 Raymond E Goldstein, University of Cambridge, United Kingdom
Publication history
 Received: May 5, 2020
 Accepted: February 12, 2021
 Version of Record published: April 26, 2021 (version 1)
 Version of Record updated: June 22, 2021 (version 2)
Copyright
© 2021, Cicconofri et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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